On Conjugate Convex Functions

Abstract

Since the classical work of Minkowski and Jensen it is well known that many of the inequalities used in analysis may be considered as consequences of the convexity of certain functions. In several of these inequalities pairs of "conjugate" functions occur, for instance pairs of powers with exponents a and a related by 1/a + 1/a = 1. A more general example is the pair of positively homogeneous convex functions defined by Minkowski and known as the distance (or gauge) function and the function of support of a convex body. The purpose of the present paper is to explain the general (by the way rather elementary) idea underlying this correspondence.